- Email: [email protected]
Alternative stochastic specifications of the frontier production function in the analysis of agricultural credit programs and technical efficiency
Journal of Development Economics 21 (1986) 149-160. North-Holland
ALTERNATIVE STOCHASTIC SPECIFICATIONS OF THE FRONTIER PRODUCTION FUNCTION IN THE ANALYSIS OF AGRICULTURAL CREDIT PROGRAMS AND TECHNICAL EFFICIENCY Timothy G. TAYLOR and J. Scott SHONKWlLER University of Florida, Gainesville, FL32611, USA Received October 1984, final version received December 1984 The effect of subsidized credit on the technical efficiency of traditional farmers in Southeastern Brazil is analyzed under two alternative stochastic specifications for the production frontier. It is found that the choice of stochastic specification significantly influences inferences regarding the effect of subsidized credit on measured technical efficiency.
1. Introduction
The provision of agricultural credit at subsidized rates of interest has a long history in Brazil as a policy action directed towards improving the productivity of traditional farmers [Adams (1971), Araujo and Meyer (1978)]. In spite of the number and variations of such programs, there remains, however, a considerable lack of consensus regarding the effectiveness and merit of such programs. Considerable research analyzing traditional Brazilian farming in light of the potential and actual effectiveness of subsidized credit programs has led to this lack of consensus [Rao (1970), Nelson (1971), Garcia (1975), Drummond (1972), Graber (1976), Teixeira (1976)]. Generally these research efforts have to some degree taken the 'poor but efficient' hypothesis [Shultz (1964)] as a starting point and analyzed the allocative efficiency of traditional farmers, rendering conclusions on this aspect of the production process alone. As noted by Steitieh (1971), however, there is a second dimension to the effectiveness of these programs. In analyzing traditional agriculture in Southern Brazil he concluded that 'increased investment in inputs (capital formulation), such as mechanized equipment and fertilizer alone is not the answer to increasing crop production. Better management, information and utilization are as important and should be equally emphasized if any benefit is to be expected from increasing expenditure on these inputs' (p. 96). 0304-3878/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
150
T.G. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
An implication of Steitieh's conclusion is that while subsidized credit availability may afford traditional farmers the opportunity to invest in modernized inputs, there is no guarantee that these inputs will be used in such manner as to realize the full extent of output gains possible. Thus, one measure of the effectiveness of subsidized credit programs is the increase in technical efficiency traditional farmers are able to achieve. The goal of this paper is to evaluate the effects of a recent World Bank sponsored credit program (PRODEMATA) in the Zona da Mata region of Brazil. In particular, we seek to evaluate the technical efficiency of program participants vis-a-vis a comparable-group of non-participants. Technical efficiency measures are obtained within the framework of econometric frontier production functions. However, there are two standard forms of the frontier production function: the full frontier and stochastic frontier. Each specification of the production frontier can yield different measures of technical efficiency, yet there is little empirical evidence to guide in the selection of one approach over the other. Thus, a second goal of the paper is to investigate the robustness of inferences concerning the contribution of PRODEMATA toward improving the technical efficiency of traditional farmers under two different stochastic specifications for the frontier production function. The plan of the paper is as follows. The second section presents the general theoretical and empirical concepts underlying the full and stochastic frontier production function specifications used in the paper. The data, empirical models and estimators are introduced in the third section. The fourth section discusses the empirical results, with the final section presenting conclusions. 2. Full and stochastic frontiers
The notion of technical efficiency in production and frontier production functions I is directly related to the theoretical definition of a production function as a mathematical form yielding the maximum output attainable from any given set of inputs. From this definition, the frontier production function represents an upper bound on output. Given a sample of firms sharing a common technology as embodied in the frontier function, the observed outputs of these firms can lie on or below the frontier but not above it. The degree to which any given firm's output falls short of the frontier output provides a logical measure of technical inefficiency. Within the context of the present analysis, if the PRODEMATA program was successful in improving the technical and managerial abilities of traditional 1Although the discussion here is concerned only with frontier production functions, frontier cost and profit functions may also be defined in a dual context.
T.G. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
151
farmers, the measured technical efficiencies of participant farmers should on average exceed those of farmers who did not participate in the program. Although Farrell (1957) was the first to provide a framework for the computation of production frontiers and efficiency measures, Aigner and Chu (1968) first explicitly specified a parametric form of the production function for the frontier. From these beginnings there has developed a diversity of frontier models [see Forsund, Lovell and Schmidt (1980)]. However, from a theoretical and econometric viewpoint these models may be placed into two broad categories, full frontier models and stochastic frontier models. Both the full and stochastic frontier production function models yield measures of estimated technical efficiency. However, they differ considerably as to the factors which are considered as contributing to measured technical inefficiency. It is not then implausible that inferences regarding technical efficiency obtained from these two models may vary. The first statistical full frontier model was proposed by Afriat (1972). The basic structure of Afriat's model and subsequent full frontier formulations may be expressed as
Y = f(x; fl)e -u,
(1)
where f(x; fl) denotes the frontier production function and u is a one sided non-negative disturbance. 2 It is immediate that the constraint u > 0 in eq. (1) ensures that the observed output of any firm necessarily lies on or below the estimated frontier. While the presence of a one-sided disturbance ensures the estimated frontier is fully consistent with the theoretical notion of a production function, it does lead to some question concerning what components should contribute to measured technical efficiency. As specified, estimation of the full frontier model from a random sample of firms implicitly assumes that all random variation is attributable to technical inefficiency. Thus, random variations in output attributable to factors exogeneous to the firm, e.g. weather, may contribute to measured technical inefficiency. Questions concerning whether or not factors outside the control of the firm and general statistical 'noise' should be properly considered as contributing to technical inefficiency led to the development of the stochastic frontier [Aigner, Lovell and Schmidt (1977), Meeusen and Van den Broeck (1977)-]. The general form of the stochastic frontier production function 3 is 2Various distributional assumptions have been proposed for u in eq. (1). Some examples include two parameter beta [Afriat (1972)], gamma [Greene (1980)], exponential and half normal [Schmidt (1976)]. 3It is interesting to note, that the stochastic specification in eq. (2) may be interpreted in terms of the general and specific ignorance which Marchak and Andrews (1944), and later Zellner, Kmenta and Dreze (1966), considered as comprising the stochastic disturbance of a production relation.
152
T.G. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
Y = f(x; fl)e v-',
(2)
where v is characterized by a symmetric distribution and u, as in the full frontier model, is a one-sided non-negative disturbance. In constrast to the full frontier specification in which the frontier If(x; fl)] is deterministic, the frontier in eq. (2) is stochastic, being defined by f(x; fl)e v. The stochastic nature of the frontier is directly related to the possible existence of random factors outside the finn's control. In contrast to the full frontier which counts such random variation towards measured technical inefficiency, the stochastic frontier specification includes a symmetric disturbance component to account for such variations. Technical inefficiency endogenous to the firm is then captured by the one-sided disturbance appended to this symmetric random variable. Given these two specifications, the impact of the PRODEMATA program on the technical efficiency of traditional farmers can be evaluated by using two theoretically valid, but rather significantly different stochastic specifications for the frontier production functions. In general, given the stochastic specifications for each model, unless the influence of random factors on production are negligible, measures of technical efficiency obtained from the stochastic frontier specification can be expected to be higher than those obtained from the full frontier model. However, whether the relative technical efficiencies between participant and non-participant farms are similar across model specifications cannot be ascertained a priori. This is a significant issue. For if competing specifications yield conflicting inferences regarding the effectiveness of the PRODEMATA program, the choice of which inference is appropriate must rest on the researcher's subjective beliefs concerning the frontier function's stochastic specification.
3. Model specification and estimation The PRODEMATA program was initiated in 1976 with the broad objective of inducing agricultural development in the Zona da Mata region of Brazil. The major instruments to achieve this goal were the provision of agricultural credit and technical services, including extension activities and the construction of research and demonstration facilities. Because the extension and technical assistance dimensions of the program were directed toward improving the technical and managerial abilities of participant farmers, a comparison of technical efficiencies between representative groups of participant and non-participant farmers should provide a good indication of the effectiveness of the PRODEMATA program in attaining some of its major goals. To assess the effect of the PRODEMATA program on the technical efficiency of traditional farmers in the Zona da Mata region of Brazil direct
TG. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
153
estimation of both a full and stochastic frontier production function was undertaken. 4 This permits a comparison of measured technical inefficiency between these two alternative specifications and provides some insight as to the robustness of inferences regarding the effects of PRODEMATA on technical efficiency to the stochastic specification of the production frontier. The data utilized in the analysis were comprised of a cross-section of 433 farm firms for the 1981-82 crop season. These data were collected as part of the PRODEMATA program by the Agricultural Economics Department of the Federal University of Vicosa. To assess the impact of PRODEMATA on technical efficiency, the data were partitioned on the basis of participation in the program. The non-participant group was composed of 252 farms that had never participated in the program. The participant group was composed of 181 farms that had participated in the PRODEMATA for at least one year. The basic form of the production function for each group is defined by 5 In Y~= In A + B~ In T~+ B 2 In Li + B3In M~,
i = 1,..., N i,
(3)
where Y~ denotes output, T~ is land in production, Li is a measure of labor, Mi denotes intermediate materials, and Nj notes the number of observations in the participant, j = P , and non-participant, j = N P , groups. Output is measured as the gross value of output deflated by the index of prices received for the Minas Gerais state (FGV). Land is measured in hectares of land utilized in agricultural production. Labor is measured in man-day equivalents, and intermediate materials are defined as the deflated value of expenditures on seed, fertilizers, pesticides, draft animal services and mechanized services. The deflator used was the index of price paid for production items in Minas Gerais (FGV). The full frontier production function is estimated by assuming that the disturbance component is distributed as a gamma random variable. Rewriting eq. (3) in obvious matrix notation yields a model of the form Y i =- X i f l - - Ui,
i = 1, •.., N~,
(4)
where Yi denotes the ith observation on the logarithm of output, and xi is vector composed of the logarithms of the regressors in (3) and ui is assumed to be independent and identically distributed as a gamma random variable independent of xi. Note that the constant term is subsumed in the parameter vector fl with xi being suitably modified. 4In order to statistically justify single equation estimation of the production frontiers, optimizing behavior is assumed to be characterized by expected profit maximization [Zellner, Kmenta and Dreze (1966)]. 5The stochastic specification of the production frontiers will be considered in subsequent discussion.
154
T.G. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
The choice of a gamma distribution for the efficiency distribution is necessary to ensure that consistent and asymptotically efficient estimates of the frontier parameters may be obtained. As noted by Schmidt, other distributional assumptions can be utilized to compute maximum likelihood (ML) 'estimates', but the resulting 'estimates' do not possess well defined standard errors because a fundamental regularity condition for obtaining valid estimates of standard errors is violated. 6 Greene (1980), however, has shown that if u ~ G ( 2 , P ) with 2 > 0 and P > 2 , the ML estimation yields consistent and asymptotically efficient parameter estimates. Estimation of the full frontier production function was achieved using maximum likelihood. The log likelihood equation for the full frontier production function was maximized for the participant and non-participant groups using a modified scoring algorithm proposed by Greene (1980). Starting values for the parameters were obtained from modified corrected ordinary least squares (COLS) estimates. 7 The stochastic frontier specification may be written as
yi=xi~i+Si,
(5)
where the disturbance component is defined as ei= v i - u i with vi"~iidN(o, a~) and ui~iid N ( o , a 2) . Furthermore vi and ui are assumed to be uncorrelated and e~ is assumed to be independent of x~. Estimation of the parameters of the stochastic frontier was accomplished by maximum likelihood. Maximization of the log likelihood function for the stochastic frontier production function was accomplished by an iterated ordinary least squares algorithm developed by Greene (1982) and modified by Lee (1983). This algorithm is based on an iterative solution to the likelihood equations and yields the appropriate ML estimates. Standard errors for the parameter estimates were obtained following Greene (1982, p. 287). Starting values used were OLS parameter estimates of the production function. 8
4. Empirical results Ordinary least squares (OLS) and maximum likelihood parameter estimates for the full and stochastic frontier production functions are presented in table 1. For the participant group, the estimated output elasticities are very similar for both the stochastic frontier and fitted average function obtained 6More precisely, the range of the distribution is a function of the parameters being estimated, violating a fundamental condition necessary to obtain the sampling the properties of the parameter estimates. 7A complete discussion of the modified COLS estimator and obtaining starting values may be found in Green (1980). 8The OLS estimate of the intercept was adjusted following Greene (1982, p. 288).
T.G. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
155
Table 1 Estimated full and stochastic frontier production parameters. Variables Estimator OLS: Participant Non-participant MLFF: b Participant Non-participant MLSF: c Participant Non-participant
Variables
Intercept
Land
Labor
Materials
try2
1.8178 (0.3377)" 1.9076 (0.2153)
0.0873 (0.0503) 0.0906 (0.0365)
0.6670 (0.0942) 0.6445 (0.0595)
0.2866 (0.0539) 0.2799 (0.0338)
0.2252 (--) 0.2453 (--)
3.3496 (--) 4.9950 (--)
0.0481 (0.0448) 0.0957 (0.0301)
0.7948 (0.0587) 0.6597 (0.0342)
0.2535 (0.0512) 0.2523 (0.0338)
1.9752 (0.3707) 2.0723 (0.2592)
0.0869 (0.0463) 0.0910 (0.0346)
0.6681 (0.0860) 0.6446 (0.0500)
0.2860 (0.0469) 0.2788 (0.0264)
tr~2
0.2344
(--)
0.2599
(--) 0.0394 (0.0877) 0.0401 (0.0825)
0.2059 (0.0458) 0.2268 (0.0374)
aAsymptotic standard errors in parentheses. bMaximum likelihood full frontier. CMaximum likelihood stochastic frontier.
by OLS. Estimated output elasticities were approximately 0.09, 0.67 and 0.29 for land, labor and materials, respectively, under both specifications. In contrast, under the full frontier specification, the estimated output elasticities for land, labor and materials were about 0.05, 0.80 and 0.25, respectively. The divergence in output elasticities between the full frontier and stochastic frontier is possibly due to the degree of skewness present in the gamma distribution assumed for the full frontier disturbance. In general, the degree of skewness of this distribution partially determines whether or not the estimated frontier is a neutrally or non-neutrally scaled version of the fitted average function. 9 As the distribution becomes more skewed, the output elasticities between the average and frontier function will tend to diverge. For the non-participant group the output elasticities for all inputs were very similar under all three specifications. The output elasticity for land ranged between 0.09 and 0.10 while those for labor and materials ranged between 0.64 to 0.66 and 0.25 to 0.28, respectively. It is interesting to note that with the exception of the full frontier for the participant group, the estimated output elasticities were very similar for both participants and nonparticipants. This provides an indication that any effects which the 9In the present context, neutral scaling would imply that the output elasticities of both the fitted average and frontier functions would be similar with the majority of difference being in the estimated intercepts.
T.G. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
156
PRODEMATA had on technical efficiency were neutral in regard to input usage. Estimated technical efficiency measures on the basis of participation and farm size are presented in table 2. The efficiency measures corresponding to the full frontier specification are Farrell measures I-Kopp (1981)]. Technical efficiency measures for individual firms based on the estimated stochastic frontiers are obtained by using the conditional expectation of ui given ~i,E(ui[ei) [see eq. (5)]. Jondrow et al. (1981) have demonstrated the conditional distribution of ui given e~ is that of an N(#i, o"2) random variable 2 2 2 2 2 2 truncated at zero with #~=-0.,,eft0. and 0..=a,,0.v/0. where 0 . 2 - 0.,,2 +0.2. An estimate of technical efficiency for the ith firm is then 1 - E(u~[ei), i = 1,..., N~. As expected, for both participant and non-participant groups, technical efficiency estimates obtained from the stochastic frontier specification are uniformly higher than those obtained from the full frontier specification. The magnitude of the difference is, perhaps, somewhat surprising. For the participant group, the average measured technical efficiency for all farms increased almost four-fold from 0.185 for the full frontier to 0.714 obtained from the stochastic frontier specification. Similarly, for the non-participant group, measured technical efficiency for all farms increased nearly twelvefold, from 0.059 under the full frontier specification to 0.704 for the stochastic frontier. A plausible explanation for the much larger increase in measured technical efficiency of the non-participant group across the two specifications as opposed to the participant group may perhaps be related to the extension Table 2 Estimated technical efficiency by farm size and participation under full and stochastic frontier specifications. Farm size (hectares) Group specification Participant: Full frontier Stochastic frontier N
Non-participant Full frontier Stochastic frontier N
< 10
10.01-50
50.01-100
> 100
All farms
0.179 (0.012) a 0.740 (0.039)
0.175 (0.007) 0.683 (0.025)
0.196 (0.014) 0.777 (0.034)
0.202 (0.021) 0.758 (0.070)
0.185 (0.006) 0.714 (0.018)
34
105
25
15
181
0.050 (0.004) 0.708 (0.023) 129
0.060 (0.009) 0.691 (0.024) 87
0.053 (0.005) 0.720 (0.043) 26
0.056 (0.007) 0.739 (0.071) 10
0.059 (0.003) 0.704 (0.015) 252
aEstimated standard errors in parentheses.
T.G. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
157
and technical assistance dimensions of the PRODEMATA program. The effect of these program features could be such that the participant farmers' production practices are considerably more homogeneous than those of nonparticipant farmers. Furthermore, it might be expected that measurement errors in the data for participant farmers would be somewhat smaller than for the non-participant group given the former group's much closer association with program personnel. Thus, data for the non-participant group probably exhibit much greater random variation and measurement error. Under the full frontier specification, all variation directly contributes to measured technical inefficiency. Thus, under this specification, it is perhaps not surprising that measured technical efficiency for non-participants is considerably smaller than that for participants. It could thus be argued that under the full frontier specification, efficiency measures are artificially low due to the presence of higher random variation and measurement errors. The effect of the PRODEMATA program on technical efficiency in production can be gauged by comparing estimated technical efficiencies between the participant and non-participant groups. If the participant group has a higher level of technical efficiency relative to its frontier than the nonparticipant group, then it is plausible that the PRODEMATA program had a significant impact on technical efficiency. For each stochastic specification, table 3 contains the ratio of average technical efficiency of participant farms to that of non-participant farms for various farm sizes. The ratios indicate that across all farm sizes, participant farms exhibited estimated technical efficiencies considerably greater than the non-participant group under the full frontier specification. Estimated technical efficiency ratios varied from 2.9 for farms in the 10 to 50 hectares category to 3.7 for farms in the 50 to 100 hectares category. The estimated efficiency ratio for all farms was approximately 3.1. Based on the results of the full frontier specification, it appears that the PRODEMATA program had a substantial impact on the technical efficiency of traditional farmers. Participant farmers were roughly 3 times more efficient relative to their frontier than non-participant farmers. Further, viewing the
Table 3 Ratios of estimated technical efficiency of participants to non-participants by farm size. Farm size (hectares) Specification
< 10
10.01-50
50.01-100
> 100
All farms
Full frontier Stochastic frontier
3.58
2.90
3.70
3.60
3.13
1.05
0.99
1.07
1.03
1.01
J.D.E.-- F
158
T.G. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
average technical efficiencies in table 2 in conjunction with the efficiency ratios in table 3, it is apparent that measured increases in technical efficiency were relatively uniform across farm sizes. The PRODEMATA program can thus be viewed as not being biased toward larger farms as is often the case when such subsidized credit programs are enacted. The picture concerning the effectiveness of the PRODEMATA program on technical efficiency that emerges under the stochastic frontier specification is, however, considerably different. The efficiency ratios in table 3 for the stochastic frontier specification indicate that participant and non-participant farms exhibited virtually equal technical efficiency relative to the respective frontiers. The lowest estimated efficiency ratio was 0.99 for farms in the 10 to 50 hectare range while the highest efficiency ratio was 1.07 for farms in the 50 to 100 hectares category. The efficiency ratio for all farms was 1.01. On the basis of the technical efficiency measures obtained from the stochastic frontier the PRODEMATA program appears to have had virtually no impact on the technical efficiency of traditional farmers. It is, however, interesting to note that estimated technical efficiencies across farm sizes for both participant and non-participant groups were fairly uniform. 5. Conclusions
Under the two alternative frontier specifications considered, quite different inferences can be formulated. Based on the full frontier specification, the empirical results indicate that the PRODEMATA program had a rather substantial impact on technical efficiencies as participant farmers exhibited technical efficiencies more than three times greater than non-participant farmers. In contrast, under the stochastic frontier specification, both participant and non-participant farmers exhibited very similar estimated technical efficiencies implying that PRODEMATA had virtually no impact on technical efficiency. The divergence leads to the rather difficult questions of which inference does one believe. The stochastic specification for the full frontier is somewhat troubling. The presence of a single one-sided disturbance necessarily implies that all variation in the sample is considered as contributing to measured technical efficiency. Thus, measurement error, and other random variation in addition to those factors under the firm's control affect estimated efficiencies. In the present analysis, it appears that the effects of such measurement errors and random variation lead to a substantial underestimation of technical efficiencies under the full frontier specification. That the estimated technical efficiencies obtained from the stochastic frontier were higher than those obtained from the full frontier is not surprising. However, the magnitude of the difference (four-fold for participants, twelve-fold for nonparticipants) taken in conjunction with the extremely low efficiency estimates
T.G. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
159
implied by the full frontier specification, cast considerable doubt as to the usefulness of this specification and hence the validity of inferences derived from it. Thus, an appropriate conclusion regarding this analysis is that the stochastic frontier specification yields little effect on the technical efficiencies of participant farmers. It is somewhat disappointing that the two alternative specifications yield conflicting inferences regarding the effectiveness of the PRODEMATA program in improving technical efficiency. Such results place considerable importance on the subjective beliefs of the researcher. Any definitive inference must, in the end, rest in the grey area of determing which specification is the most realistic on both theoretical and empirical grounds. References Adams, D.W., 1971, Agricultural credit in Latin America: A critical appraisal of external funding policy, American Journal of Agricultural Economics 53, 163--172. Afriat, S.N., 1972, Efficiency estimation of production functions, International Economic Review 13, 568-598. Aigner, D.J. and D.S. Chu., 1968, On estimating the industry production function, American Economic Review 58, 826--839. Aigner, D.J., K. Lovell and P. Schmidt, 1977, Formulation and estimation of stochastic frontier production function models, Journal of Econometrics 5, 21-38. Araujo, P. and R.L. Meyer, 1978, Agricultural credit policy in Brazil, Objectives and results, Savings and Development 3, 169-172. Drummond, H.E., 1972, An economic analysis of farm enterprise diversifications and associated factors in two regions of Midas GerMs, Brazil, Unpublished Ph.D. dissertation (Purdue University, West Lafayette, IN). Fan'ell, M.J., 1957, The measurement of productive efficiency, Journal of the Royal Statistical Society, Series A, 253-281. Forsund, F.A., C.A.K. Lovell and P. Schmidt, 1980, A survey of frontier production functions and of their relationship to efficiency measurement, Journal of Econometrics 13, 5-25. Fundac~o Getulio Vargas (FGV), Centro de Estudos Agricolos, 1982. Garcia, J.C., 1975, Analise de alocafao de recursos for proprietarion e parceiros em areas de agricultura de subsistencia, Unpublished M.S. thesis (Universidade Federal de Vi~osa, Viqosa). Graber, K.L., 1976, Factors explaining farm production and family earnings of small farmers in Brazil, Unpublished Ph.D. dissertation (Purdue University, West Lafayette, IN). Greene, W.H., 1980, Maximum likelihood estimation of econometric frontier functions, Journal of Econometrics 13, 27-56. Greene, W.H., 1982, Maximum likelihood estimation of stochastic frontier production models, Journal of Econometrics 18, 285-290. Jondrow, J., C.A.K. Lovell, I.S. Materov and P. Schmidt, 1982, On the estimation of technical inefficiency in the stochastic frontier production function model, Journal of Econometrics 19, 233-238. Kopp, R.J., 1981, The measurement of productive efficiency: A reconsideration, Quarterly Journal of Economics 96, 477-503. Lee, L.F., 1983, On maximum likelihood estimation of stochastic frontier production models, Journal of Econometrics 23, 269-274. Marchak, J. and WJ. Andrews, 1944, Random simultaneous equations and the theory of production, Econometrica 12, 143-205. Meeusen, W. and J. van den Broeck, 1977, Efficiency estimation from Cobb-Douglas production functions with composed error, International Economic Review 18, 435-444. Nelson, W.C., 1971, An economic analysis of fertilizer utilization in Brazil, Unpublished Ph.D. dissertation (Ohio State University, Columbus, OH).
160
T.G. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
Rao, B.P., 1970, The economics of agricultural credit use in Brazil, Unpublished Ph.D. dissertation (Ohio State University, Columbus, OH). Schmidt, P., 1976, On the statistical estimation of parametric frontier production functions, Review of Economics and Statistics 58, 238-239. Shultz, T.W., 1964, Transforming traditional agriculture (Yale University Press, New Haven, CT). Steitieh, A.M., 1971, Input productivity and productivity change of crop enterprise in Southern Brazil, Unpublished Ph.D. dissertation (Ohio State University, Columbus, OH). Teixeira, T.D., 1976, Resource efficiency and the market for family labor: Small farms in the Sertao of Northeast Brazil, Unpublisheded Ph.D. dissertation (Purdue University, West Lafayette, IN). Zellner, A., J. Kmenta and J. Dreze, 1966, Specification and estimation of Cobb-Douglas production function models, Econometrica 34, 784-795.
ALTERNATIVE STOCHASTIC SPECIFICATIONS OF THE FRONTIER PRODUCTION FUNCTION IN THE ANALYSIS OF AGRICULTURAL CREDIT PROGRAMS AND TECHNICAL EFFICIENCY Timothy G. TAYLOR and J. Scott SHONKWlLER University of Florida, Gainesville, FL32611, USA Received October 1984, final version received December 1984 The effect of subsidized credit on the technical efficiency of traditional farmers in Southeastern Brazil is analyzed under two alternative stochastic specifications for the production frontier. It is found that the choice of stochastic specification significantly influences inferences regarding the effect of subsidized credit on measured technical efficiency.
1. Introduction
The provision of agricultural credit at subsidized rates of interest has a long history in Brazil as a policy action directed towards improving the productivity of traditional farmers [Adams (1971), Araujo and Meyer (1978)]. In spite of the number and variations of such programs, there remains, however, a considerable lack of consensus regarding the effectiveness and merit of such programs. Considerable research analyzing traditional Brazilian farming in light of the potential and actual effectiveness of subsidized credit programs has led to this lack of consensus [Rao (1970), Nelson (1971), Garcia (1975), Drummond (1972), Graber (1976), Teixeira (1976)]. Generally these research efforts have to some degree taken the 'poor but efficient' hypothesis [Shultz (1964)] as a starting point and analyzed the allocative efficiency of traditional farmers, rendering conclusions on this aspect of the production process alone. As noted by Steitieh (1971), however, there is a second dimension to the effectiveness of these programs. In analyzing traditional agriculture in Southern Brazil he concluded that 'increased investment in inputs (capital formulation), such as mechanized equipment and fertilizer alone is not the answer to increasing crop production. Better management, information and utilization are as important and should be equally emphasized if any benefit is to be expected from increasing expenditure on these inputs' (p. 96). 0304-3878/86/$3.50 © 1986, Elsevier Science Publishers B.V. (North-Holland)
150
T.G. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
An implication of Steitieh's conclusion is that while subsidized credit availability may afford traditional farmers the opportunity to invest in modernized inputs, there is no guarantee that these inputs will be used in such manner as to realize the full extent of output gains possible. Thus, one measure of the effectiveness of subsidized credit programs is the increase in technical efficiency traditional farmers are able to achieve. The goal of this paper is to evaluate the effects of a recent World Bank sponsored credit program (PRODEMATA) in the Zona da Mata region of Brazil. In particular, we seek to evaluate the technical efficiency of program participants vis-a-vis a comparable-group of non-participants. Technical efficiency measures are obtained within the framework of econometric frontier production functions. However, there are two standard forms of the frontier production function: the full frontier and stochastic frontier. Each specification of the production frontier can yield different measures of technical efficiency, yet there is little empirical evidence to guide in the selection of one approach over the other. Thus, a second goal of the paper is to investigate the robustness of inferences concerning the contribution of PRODEMATA toward improving the technical efficiency of traditional farmers under two different stochastic specifications for the frontier production function. The plan of the paper is as follows. The second section presents the general theoretical and empirical concepts underlying the full and stochastic frontier production function specifications used in the paper. The data, empirical models and estimators are introduced in the third section. The fourth section discusses the empirical results, with the final section presenting conclusions. 2. Full and stochastic frontiers
The notion of technical efficiency in production and frontier production functions I is directly related to the theoretical definition of a production function as a mathematical form yielding the maximum output attainable from any given set of inputs. From this definition, the frontier production function represents an upper bound on output. Given a sample of firms sharing a common technology as embodied in the frontier function, the observed outputs of these firms can lie on or below the frontier but not above it. The degree to which any given firm's output falls short of the frontier output provides a logical measure of technical inefficiency. Within the context of the present analysis, if the PRODEMATA program was successful in improving the technical and managerial abilities of traditional 1Although the discussion here is concerned only with frontier production functions, frontier cost and profit functions may also be defined in a dual context.
T.G. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
151
farmers, the measured technical efficiencies of participant farmers should on average exceed those of farmers who did not participate in the program. Although Farrell (1957) was the first to provide a framework for the computation of production frontiers and efficiency measures, Aigner and Chu (1968) first explicitly specified a parametric form of the production function for the frontier. From these beginnings there has developed a diversity of frontier models [see Forsund, Lovell and Schmidt (1980)]. However, from a theoretical and econometric viewpoint these models may be placed into two broad categories, full frontier models and stochastic frontier models. Both the full and stochastic frontier production function models yield measures of estimated technical efficiency. However, they differ considerably as to the factors which are considered as contributing to measured technical inefficiency. It is not then implausible that inferences regarding technical efficiency obtained from these two models may vary. The first statistical full frontier model was proposed by Afriat (1972). The basic structure of Afriat's model and subsequent full frontier formulations may be expressed as
Y = f(x; fl)e -u,
(1)
where f(x; fl) denotes the frontier production function and u is a one sided non-negative disturbance. 2 It is immediate that the constraint u > 0 in eq. (1) ensures that the observed output of any firm necessarily lies on or below the estimated frontier. While the presence of a one-sided disturbance ensures the estimated frontier is fully consistent with the theoretical notion of a production function, it does lead to some question concerning what components should contribute to measured technical efficiency. As specified, estimation of the full frontier model from a random sample of firms implicitly assumes that all random variation is attributable to technical inefficiency. Thus, random variations in output attributable to factors exogeneous to the firm, e.g. weather, may contribute to measured technical inefficiency. Questions concerning whether or not factors outside the control of the firm and general statistical 'noise' should be properly considered as contributing to technical inefficiency led to the development of the stochastic frontier [Aigner, Lovell and Schmidt (1977), Meeusen and Van den Broeck (1977)-]. The general form of the stochastic frontier production function 3 is 2Various distributional assumptions have been proposed for u in eq. (1). Some examples include two parameter beta [Afriat (1972)], gamma [Greene (1980)], exponential and half normal [Schmidt (1976)]. 3It is interesting to note, that the stochastic specification in eq. (2) may be interpreted in terms of the general and specific ignorance which Marchak and Andrews (1944), and later Zellner, Kmenta and Dreze (1966), considered as comprising the stochastic disturbance of a production relation.
152
T.G. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
Y = f(x; fl)e v-',
(2)
where v is characterized by a symmetric distribution and u, as in the full frontier model, is a one-sided non-negative disturbance. In constrast to the full frontier specification in which the frontier If(x; fl)] is deterministic, the frontier in eq. (2) is stochastic, being defined by f(x; fl)e v. The stochastic nature of the frontier is directly related to the possible existence of random factors outside the finn's control. In contrast to the full frontier which counts such random variation towards measured technical inefficiency, the stochastic frontier specification includes a symmetric disturbance component to account for such variations. Technical inefficiency endogenous to the firm is then captured by the one-sided disturbance appended to this symmetric random variable. Given these two specifications, the impact of the PRODEMATA program on the technical efficiency of traditional farmers can be evaluated by using two theoretically valid, but rather significantly different stochastic specifications for the frontier production functions. In general, given the stochastic specifications for each model, unless the influence of random factors on production are negligible, measures of technical efficiency obtained from the stochastic frontier specification can be expected to be higher than those obtained from the full frontier model. However, whether the relative technical efficiencies between participant and non-participant farms are similar across model specifications cannot be ascertained a priori. This is a significant issue. For if competing specifications yield conflicting inferences regarding the effectiveness of the PRODEMATA program, the choice of which inference is appropriate must rest on the researcher's subjective beliefs concerning the frontier function's stochastic specification.
3. Model specification and estimation The PRODEMATA program was initiated in 1976 with the broad objective of inducing agricultural development in the Zona da Mata region of Brazil. The major instruments to achieve this goal were the provision of agricultural credit and technical services, including extension activities and the construction of research and demonstration facilities. Because the extension and technical assistance dimensions of the program were directed toward improving the technical and managerial abilities of participant farmers, a comparison of technical efficiencies between representative groups of participant and non-participant farmers should provide a good indication of the effectiveness of the PRODEMATA program in attaining some of its major goals. To assess the effect of the PRODEMATA program on the technical efficiency of traditional farmers in the Zona da Mata region of Brazil direct
TG. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
153
estimation of both a full and stochastic frontier production function was undertaken. 4 This permits a comparison of measured technical inefficiency between these two alternative specifications and provides some insight as to the robustness of inferences regarding the effects of PRODEMATA on technical efficiency to the stochastic specification of the production frontier. The data utilized in the analysis were comprised of a cross-section of 433 farm firms for the 1981-82 crop season. These data were collected as part of the PRODEMATA program by the Agricultural Economics Department of the Federal University of Vicosa. To assess the impact of PRODEMATA on technical efficiency, the data were partitioned on the basis of participation in the program. The non-participant group was composed of 252 farms that had never participated in the program. The participant group was composed of 181 farms that had participated in the PRODEMATA for at least one year. The basic form of the production function for each group is defined by 5 In Y~= In A + B~ In T~+ B 2 In Li + B3In M~,
i = 1,..., N i,
(3)
where Y~ denotes output, T~ is land in production, Li is a measure of labor, Mi denotes intermediate materials, and Nj notes the number of observations in the participant, j = P , and non-participant, j = N P , groups. Output is measured as the gross value of output deflated by the index of prices received for the Minas Gerais state (FGV). Land is measured in hectares of land utilized in agricultural production. Labor is measured in man-day equivalents, and intermediate materials are defined as the deflated value of expenditures on seed, fertilizers, pesticides, draft animal services and mechanized services. The deflator used was the index of price paid for production items in Minas Gerais (FGV). The full frontier production function is estimated by assuming that the disturbance component is distributed as a gamma random variable. Rewriting eq. (3) in obvious matrix notation yields a model of the form Y i =- X i f l - - Ui,
i = 1, •.., N~,
(4)
where Yi denotes the ith observation on the logarithm of output, and xi is vector composed of the logarithms of the regressors in (3) and ui is assumed to be independent and identically distributed as a gamma random variable independent of xi. Note that the constant term is subsumed in the parameter vector fl with xi being suitably modified. 4In order to statistically justify single equation estimation of the production frontiers, optimizing behavior is assumed to be characterized by expected profit maximization [Zellner, Kmenta and Dreze (1966)]. 5The stochastic specification of the production frontiers will be considered in subsequent discussion.
154
T.G. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
The choice of a gamma distribution for the efficiency distribution is necessary to ensure that consistent and asymptotically efficient estimates of the frontier parameters may be obtained. As noted by Schmidt, other distributional assumptions can be utilized to compute maximum likelihood (ML) 'estimates', but the resulting 'estimates' do not possess well defined standard errors because a fundamental regularity condition for obtaining valid estimates of standard errors is violated. 6 Greene (1980), however, has shown that if u ~ G ( 2 , P ) with 2 > 0 and P > 2 , the ML estimation yields consistent and asymptotically efficient parameter estimates. Estimation of the full frontier production function was achieved using maximum likelihood. The log likelihood equation for the full frontier production function was maximized for the participant and non-participant groups using a modified scoring algorithm proposed by Greene (1980). Starting values for the parameters were obtained from modified corrected ordinary least squares (COLS) estimates. 7 The stochastic frontier specification may be written as
yi=xi~i+Si,
(5)
where the disturbance component is defined as ei= v i - u i with vi"~iidN(o, a~) and ui~iid N ( o , a 2) . Furthermore vi and ui are assumed to be uncorrelated and e~ is assumed to be independent of x~. Estimation of the parameters of the stochastic frontier was accomplished by maximum likelihood. Maximization of the log likelihood function for the stochastic frontier production function was accomplished by an iterated ordinary least squares algorithm developed by Greene (1982) and modified by Lee (1983). This algorithm is based on an iterative solution to the likelihood equations and yields the appropriate ML estimates. Standard errors for the parameter estimates were obtained following Greene (1982, p. 287). Starting values used were OLS parameter estimates of the production function. 8
4. Empirical results Ordinary least squares (OLS) and maximum likelihood parameter estimates for the full and stochastic frontier production functions are presented in table 1. For the participant group, the estimated output elasticities are very similar for both the stochastic frontier and fitted average function obtained 6More precisely, the range of the distribution is a function of the parameters being estimated, violating a fundamental condition necessary to obtain the sampling the properties of the parameter estimates. 7A complete discussion of the modified COLS estimator and obtaining starting values may be found in Green (1980). 8The OLS estimate of the intercept was adjusted following Greene (1982, p. 288).
T.G. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
155
Table 1 Estimated full and stochastic frontier production parameters. Variables Estimator OLS: Participant Non-participant MLFF: b Participant Non-participant MLSF: c Participant Non-participant
Variables
Intercept
Land
Labor
Materials
try2
1.8178 (0.3377)" 1.9076 (0.2153)
0.0873 (0.0503) 0.0906 (0.0365)
0.6670 (0.0942) 0.6445 (0.0595)
0.2866 (0.0539) 0.2799 (0.0338)
0.2252 (--) 0.2453 (--)
3.3496 (--) 4.9950 (--)
0.0481 (0.0448) 0.0957 (0.0301)
0.7948 (0.0587) 0.6597 (0.0342)
0.2535 (0.0512) 0.2523 (0.0338)
1.9752 (0.3707) 2.0723 (0.2592)
0.0869 (0.0463) 0.0910 (0.0346)
0.6681 (0.0860) 0.6446 (0.0500)
0.2860 (0.0469) 0.2788 (0.0264)
tr~2
0.2344
(--)
0.2599
(--) 0.0394 (0.0877) 0.0401 (0.0825)
0.2059 (0.0458) 0.2268 (0.0374)
aAsymptotic standard errors in parentheses. bMaximum likelihood full frontier. CMaximum likelihood stochastic frontier.
by OLS. Estimated output elasticities were approximately 0.09, 0.67 and 0.29 for land, labor and materials, respectively, under both specifications. In contrast, under the full frontier specification, the estimated output elasticities for land, labor and materials were about 0.05, 0.80 and 0.25, respectively. The divergence in output elasticities between the full frontier and stochastic frontier is possibly due to the degree of skewness present in the gamma distribution assumed for the full frontier disturbance. In general, the degree of skewness of this distribution partially determines whether or not the estimated frontier is a neutrally or non-neutrally scaled version of the fitted average function. 9 As the distribution becomes more skewed, the output elasticities between the average and frontier function will tend to diverge. For the non-participant group the output elasticities for all inputs were very similar under all three specifications. The output elasticity for land ranged between 0.09 and 0.10 while those for labor and materials ranged between 0.64 to 0.66 and 0.25 to 0.28, respectively. It is interesting to note that with the exception of the full frontier for the participant group, the estimated output elasticities were very similar for both participants and nonparticipants. This provides an indication that any effects which the 9In the present context, neutral scaling would imply that the output elasticities of both the fitted average and frontier functions would be similar with the majority of difference being in the estimated intercepts.
T.G. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
156
PRODEMATA had on technical efficiency were neutral in regard to input usage. Estimated technical efficiency measures on the basis of participation and farm size are presented in table 2. The efficiency measures corresponding to the full frontier specification are Farrell measures I-Kopp (1981)]. Technical efficiency measures for individual firms based on the estimated stochastic frontiers are obtained by using the conditional expectation of ui given ~i,E(ui[ei) [see eq. (5)]. Jondrow et al. (1981) have demonstrated the conditional distribution of ui given e~ is that of an N(#i, o"2) random variable 2 2 2 2 2 2 truncated at zero with #~=-0.,,eft0. and 0..=a,,0.v/0. where 0 . 2 - 0.,,2 +0.2. An estimate of technical efficiency for the ith firm is then 1 - E(u~[ei), i = 1,..., N~. As expected, for both participant and non-participant groups, technical efficiency estimates obtained from the stochastic frontier specification are uniformly higher than those obtained from the full frontier specification. The magnitude of the difference is, perhaps, somewhat surprising. For the participant group, the average measured technical efficiency for all farms increased almost four-fold from 0.185 for the full frontier to 0.714 obtained from the stochastic frontier specification. Similarly, for the non-participant group, measured technical efficiency for all farms increased nearly twelvefold, from 0.059 under the full frontier specification to 0.704 for the stochastic frontier. A plausible explanation for the much larger increase in measured technical efficiency of the non-participant group across the two specifications as opposed to the participant group may perhaps be related to the extension Table 2 Estimated technical efficiency by farm size and participation under full and stochastic frontier specifications. Farm size (hectares) Group specification Participant: Full frontier Stochastic frontier N
Non-participant Full frontier Stochastic frontier N
< 10
10.01-50
50.01-100
> 100
All farms
0.179 (0.012) a 0.740 (0.039)
0.175 (0.007) 0.683 (0.025)
0.196 (0.014) 0.777 (0.034)
0.202 (0.021) 0.758 (0.070)
0.185 (0.006) 0.714 (0.018)
34
105
25
15
181
0.050 (0.004) 0.708 (0.023) 129
0.060 (0.009) 0.691 (0.024) 87
0.053 (0.005) 0.720 (0.043) 26
0.056 (0.007) 0.739 (0.071) 10
0.059 (0.003) 0.704 (0.015) 252
aEstimated standard errors in parentheses.
T.G. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
157
and technical assistance dimensions of the PRODEMATA program. The effect of these program features could be such that the participant farmers' production practices are considerably more homogeneous than those of nonparticipant farmers. Furthermore, it might be expected that measurement errors in the data for participant farmers would be somewhat smaller than for the non-participant group given the former group's much closer association with program personnel. Thus, data for the non-participant group probably exhibit much greater random variation and measurement error. Under the full frontier specification, all variation directly contributes to measured technical inefficiency. Thus, under this specification, it is perhaps not surprising that measured technical efficiency for non-participants is considerably smaller than that for participants. It could thus be argued that under the full frontier specification, efficiency measures are artificially low due to the presence of higher random variation and measurement errors. The effect of the PRODEMATA program on technical efficiency in production can be gauged by comparing estimated technical efficiencies between the participant and non-participant groups. If the participant group has a higher level of technical efficiency relative to its frontier than the nonparticipant group, then it is plausible that the PRODEMATA program had a significant impact on technical efficiency. For each stochastic specification, table 3 contains the ratio of average technical efficiency of participant farms to that of non-participant farms for various farm sizes. The ratios indicate that across all farm sizes, participant farms exhibited estimated technical efficiencies considerably greater than the non-participant group under the full frontier specification. Estimated technical efficiency ratios varied from 2.9 for farms in the 10 to 50 hectares category to 3.7 for farms in the 50 to 100 hectares category. The estimated efficiency ratio for all farms was approximately 3.1. Based on the results of the full frontier specification, it appears that the PRODEMATA program had a substantial impact on the technical efficiency of traditional farmers. Participant farmers were roughly 3 times more efficient relative to their frontier than non-participant farmers. Further, viewing the
Table 3 Ratios of estimated technical efficiency of participants to non-participants by farm size. Farm size (hectares) Specification
< 10
10.01-50
50.01-100
> 100
All farms
Full frontier Stochastic frontier
3.58
2.90
3.70
3.60
3.13
1.05
0.99
1.07
1.03
1.01
J.D.E.-- F
158
T.G. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
average technical efficiencies in table 2 in conjunction with the efficiency ratios in table 3, it is apparent that measured increases in technical efficiency were relatively uniform across farm sizes. The PRODEMATA program can thus be viewed as not being biased toward larger farms as is often the case when such subsidized credit programs are enacted. The picture concerning the effectiveness of the PRODEMATA program on technical efficiency that emerges under the stochastic frontier specification is, however, considerably different. The efficiency ratios in table 3 for the stochastic frontier specification indicate that participant and non-participant farms exhibited virtually equal technical efficiency relative to the respective frontiers. The lowest estimated efficiency ratio was 0.99 for farms in the 10 to 50 hectare range while the highest efficiency ratio was 1.07 for farms in the 50 to 100 hectares category. The efficiency ratio for all farms was 1.01. On the basis of the technical efficiency measures obtained from the stochastic frontier the PRODEMATA program appears to have had virtually no impact on the technical efficiency of traditional farmers. It is, however, interesting to note that estimated technical efficiencies across farm sizes for both participant and non-participant groups were fairly uniform. 5. Conclusions
Under the two alternative frontier specifications considered, quite different inferences can be formulated. Based on the full frontier specification, the empirical results indicate that the PRODEMATA program had a rather substantial impact on technical efficiencies as participant farmers exhibited technical efficiencies more than three times greater than non-participant farmers. In contrast, under the stochastic frontier specification, both participant and non-participant farmers exhibited very similar estimated technical efficiencies implying that PRODEMATA had virtually no impact on technical efficiency. The divergence leads to the rather difficult questions of which inference does one believe. The stochastic specification for the full frontier is somewhat troubling. The presence of a single one-sided disturbance necessarily implies that all variation in the sample is considered as contributing to measured technical efficiency. Thus, measurement error, and other random variation in addition to those factors under the firm's control affect estimated efficiencies. In the present analysis, it appears that the effects of such measurement errors and random variation lead to a substantial underestimation of technical efficiencies under the full frontier specification. That the estimated technical efficiencies obtained from the stochastic frontier were higher than those obtained from the full frontier is not surprising. However, the magnitude of the difference (four-fold for participants, twelve-fold for nonparticipants) taken in conjunction with the extremely low efficiency estimates
T.G. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
159
implied by the full frontier specification, cast considerable doubt as to the usefulness of this specification and hence the validity of inferences derived from it. Thus, an appropriate conclusion regarding this analysis is that the stochastic frontier specification yields little effect on the technical efficiencies of participant farmers. It is somewhat disappointing that the two alternative specifications yield conflicting inferences regarding the effectiveness of the PRODEMATA program in improving technical efficiency. Such results place considerable importance on the subjective beliefs of the researcher. Any definitive inference must, in the end, rest in the grey area of determing which specification is the most realistic on both theoretical and empirical grounds. References Adams, D.W., 1971, Agricultural credit in Latin America: A critical appraisal of external funding policy, American Journal of Agricultural Economics 53, 163--172. Afriat, S.N., 1972, Efficiency estimation of production functions, International Economic Review 13, 568-598. Aigner, D.J. and D.S. Chu., 1968, On estimating the industry production function, American Economic Review 58, 826--839. Aigner, D.J., K. Lovell and P. Schmidt, 1977, Formulation and estimation of stochastic frontier production function models, Journal of Econometrics 5, 21-38. Araujo, P. and R.L. Meyer, 1978, Agricultural credit policy in Brazil, Objectives and results, Savings and Development 3, 169-172. Drummond, H.E., 1972, An economic analysis of farm enterprise diversifications and associated factors in two regions of Midas GerMs, Brazil, Unpublished Ph.D. dissertation (Purdue University, West Lafayette, IN). Fan'ell, M.J., 1957, The measurement of productive efficiency, Journal of the Royal Statistical Society, Series A, 253-281. Forsund, F.A., C.A.K. Lovell and P. Schmidt, 1980, A survey of frontier production functions and of their relationship to efficiency measurement, Journal of Econometrics 13, 5-25. Fundac~o Getulio Vargas (FGV), Centro de Estudos Agricolos, 1982. Garcia, J.C., 1975, Analise de alocafao de recursos for proprietarion e parceiros em areas de agricultura de subsistencia, Unpublished M.S. thesis (Universidade Federal de Vi~osa, Viqosa). Graber, K.L., 1976, Factors explaining farm production and family earnings of small farmers in Brazil, Unpublished Ph.D. dissertation (Purdue University, West Lafayette, IN). Greene, W.H., 1980, Maximum likelihood estimation of econometric frontier functions, Journal of Econometrics 13, 27-56. Greene, W.H., 1982, Maximum likelihood estimation of stochastic frontier production models, Journal of Econometrics 18, 285-290. Jondrow, J., C.A.K. Lovell, I.S. Materov and P. Schmidt, 1982, On the estimation of technical inefficiency in the stochastic frontier production function model, Journal of Econometrics 19, 233-238. Kopp, R.J., 1981, The measurement of productive efficiency: A reconsideration, Quarterly Journal of Economics 96, 477-503. Lee, L.F., 1983, On maximum likelihood estimation of stochastic frontier production models, Journal of Econometrics 23, 269-274. Marchak, J. and WJ. Andrews, 1944, Random simultaneous equations and the theory of production, Econometrica 12, 143-205. Meeusen, W. and J. van den Broeck, 1977, Efficiency estimation from Cobb-Douglas production functions with composed error, International Economic Review 18, 435-444. Nelson, W.C., 1971, An economic analysis of fertilizer utilization in Brazil, Unpublished Ph.D. dissertation (Ohio State University, Columbus, OH).
160
T.G. Taylor and J.S. Shonkwiler, Alternative stochastic specifications...
Rao, B.P., 1970, The economics of agricultural credit use in Brazil, Unpublished Ph.D. dissertation (Ohio State University, Columbus, OH). Schmidt, P., 1976, On the statistical estimation of parametric frontier production functions, Review of Economics and Statistics 58, 238-239. Shultz, T.W., 1964, Transforming traditional agriculture (Yale University Press, New Haven, CT). Steitieh, A.M., 1971, Input productivity and productivity change of crop enterprise in Southern Brazil, Unpublished Ph.D. dissertation (Ohio State University, Columbus, OH). Teixeira, T.D., 1976, Resource efficiency and the market for family labor: Small farms in the Sertao of Northeast Brazil, Unpublisheded Ph.D. dissertation (Purdue University, West Lafayette, IN). Zellner, A., J. Kmenta and J. Dreze, 1966, Specification and estimation of Cobb-Douglas production function models, Econometrica 34, 784-795.